Linear Algebra, Spring 2011

Instructor: Andrew Obus
email: obus [at]
office: Mathematics 609
phone: 212-854-6240



MW 6:10-7:25, Mathematics 203.  Please ask questions if anything in lecture is unclear. Lectures will run the entire 75 minutes. Please show up on time!


Schedule of Lectures


Linear Algebra with Applications, 4th ed., by Otto Bretscher. This is available at the Columbia Bookstore, on, and in the Mathematics Library.


Linear Algebra is, essentially, the study of linear equations.  No doubt you have seen such equations in high school.  You probably even solved some systems of 2 or 3 simultaneous linear equations (in 2 or 3 variables).  Linear algebra takes a deeper look at such systems and examines questions such as:  How do we know when a system of m linear equations in n variables has a solution?  How many solutions can there be?  How do we find them efficiently?  If there is no solution, then how close can we get to one?  These questions may seem somewhat narrow at first glance.  But they are in fact fundamental to physics, computer science, and statistics.  Furthermore, it is only a slight stretch to say that all higher mathematics as it is practiced today (geometry, topology, number theory, analysis, differential equations, etc.) depends fundamentally on linear algebra.

The main topics we will cover are linear equations, linear transformfations, vector spaces, determinants, eigenvalues, eigenvectors, diagonalization, and applications.  These correspond to most of Chapters 1-8 of Bretscher's book.  I also hope to lecture on the Google PageRank algorithm, which is not in the book.  We will try to strike a balance between computations, concepts, proofs, and applications.  Some short proofs may appear on homeworks and exams.

Expected Background: Officially, the prerequisite is Calculus III.  In reality, we will rarely use calculus.  On the other hand, it is important that you are comfortable with vectors and basic geometry of 3-dimensional space as taught in Calculus III (vector addition, dot product, cross product, magnitude, normal vectors, etc.).

If you have any questions concerning your background, please speak to me as soon as possible.

Office Hours

Mondays 4:00-5:00, Wednesdays 10:30-11:30. Mathematics 609 (my office). If these times do not work for you, we can try to set up an appointment.


Homework will be due on Wednesdays. It is due IN CLASS. If you are going to miss class for whatever reason, you must either submit the homework to my box outside of Mathematics 410 BEFORE the beginning of class, or have a friend submit your homework in class. Late or improperly submitted homework will never be accepted. If you know in advance you will be unable to turn in homework when it is due, you should plan to turn it in ahead of time. If you have a conflict with a religious holiday, you must let me know BEFORE the fact. I will not be sympathetic if you do not tell me in advance. I will drop your lowest homework score to allow for missed assignments or for assignments that pose special difficulty.

Homework must be neat, well-organized, and legible. In addition, it must be STAPLED or PAPER CLIPPED (no folding over the top-left corner or anything like that). Please write in paragraphs, sentences, and English words (oh my!) when they are called for. The TA should not have to decipher what you are doing--you should be clear and unambiguous about your methods on a homework problem.

Homework will be graded and every effort will be made to hand it back promptly.  Grades will be posted on Courseworks.

Schedule of Homework (will also be posted on Courseworks)


Midterms will be in class on Wednesday February 16th, and Wednesday March 30th. If you have a conflict with one of these days, you must let me know now. Another exam on the same day is not considered a conflict.

The final exam is (tentatively) on Monday, May 9th, from 7:10PM-10:00PM.

Calculators are not permitted on exams.

Final Course Grades

20% Homework
20% Each Midterm
40% Final Exam

It is possible for exceptional class participation to be factored into the homework grade.

Academic Dishonesty

I will tolerate no cheating, plagiarism, or other forms of academic dishonesty. Cheating on an exam will result in an automatic grade of zero for that exam. For more serious matters, the Office of Academic Affairs may be contacted.

Some Useful Links

Calculus Placement/Information Page

Columbia Undergraduate Math Page

Columbia Math Department

Extra Help


If you have (anonymous) comments for me about teaching style or anything related to the course, click here for a feedback form.